Neural Laplace is a unified framework for learning diverse classes of differential equations (DE). For different classes of DE, this framework outperforms other approaches relying on neural networks that aim to learn classes of ordinary differential equations (ODE). However, many systems can't be modelled using ODEs. Stochastic differential equations (SDE) are the mathematical tool of choice when modelling spatiotemporal DE dynamics under the influence of randomness. In this work, we review the potential applications of Neural Laplace to learn diverse classes of SDE, both from a theoretical and a practical point of view.

Bayesian methods involve a prior belief about a model and learning from the data to arrive at a new belief, which is termed the posterior belief. Mathematically, the posterior belief can be derived from the prior belief and the empirical evidence presented by the data using Bayes' rule. In this way Bayesian analysis is a natural statistical machine learning method (see [42, 9, 33, 34, 40, 46, 30, 35] amongst many others), and especially powerful for small datasets, missing data or complex models. From a computational viewpoint, various approaches have been proposed to perform Bayesian analysis, mainly exact (analytical or sampling-based) or approximate inferential approaches. Sampling-based methods like Markov Chain Monte Carlo (MCMC) sampling with its extensions (see [28, 12, 8, 1], amongst others) gained popularity in the 1990's but suffers from slow speed and convergence issues exacerbated by large data and/or complicated models.

Several numerical approximation strategies for the expectation-propagation algorithm are studied in the context of large-scale learning: the Laplace method, a faster variant of it, Gaussian quadrature, and a deterministic version of variational sampling (i.e., combining quadrature with variational approximation). Experiments in training linear binary classifiers show that the expectation-propagation algorithm converges best using variational sampling, while it also converges well using Laplace-style methods with smooth factors but tends to be unstable with non-differentiable ones. Gaussian quadrature yields unstable behavior or convergence to a sub-optimal solution in most experiments.

We present a variational Bayesian method for model selection over families of kernels classifiers like Support Vector machines or Gaussian processes. The algorithm needs no user interaction and is able to adapt a large number of kernel parameters to given data without having to sacrifice training cases for validation. This opens the possibility to use sophisticated families of kernels in situations where the small "standard kernel" classes are clearly inappropriate. We relate the method to other work done on Gaussian processes and clarify the relation between Support Vector machines and certain Gaussian process models.

We present a variational Bayesian method for model selection over families of kernels classifiers like Support Vector machines or Gaussian processes. The algorithm needs no user interaction and is able to adapt a large number of kernel parameters to given data without having to sacrifice training cases for validation. This opens the possibility to use sophisticated families of kernels in situations where the small "standard kernel" classes are clearly inappropriate. We relate the method to other work done on Gaussian processes and clarify the relation between Support Vector machines and certain Gaussian process models.

We present a variational Bayesian method for model selection over families of kernels classifiers like Support Vector machines or Gaussian processes.The algorithm needs no user interaction and is able to adapt a large number of kernel parameters to given data without having to sacrifice training cases for validation. This opens the possibility touse sophisticated families of kernels in situations where the small "standard kernel" classes are clearly inappropriate. We relate the method to other work done on Gaussian processes and clarify the relation between Support Vector machines and certain Gaussian process models. 1 Introduction Bayesian techniques have been widely and successfully used in the neural networks and statistics community and are appealing because of their conceptual simplicity, generality and consistency with which they solve learning problems. In this paper we present a new method for applying the Bayesian methodology to Support Vector machines. We will briefly review Gaussian Process and Support Vector classification in this section and clarify their relationship by pointing out the common roots. Although we focus on classification here, it is straightforward to apply the methods to regression problems as well. In section 2 we introduce our algorithm and show relations to existing methods. Finally, we present experimental results in section 3 and close with a discussion in section 4. Let X be a measure space (e.g.